In this paper we provide a matrix extension of the scalar binomial series under elliptical contoured models and real normed division algebras. The classical hypergeometric series ${}_{1}F_{0}^{\beta}(a;\mathbf{Z})={}_{1}^{k}P_{0}^{\beta,1}(1:a;\mathbf{Z})=|\mathbf{I}-\mathbf{Z}|^{-a}$ of Jack polynomials are now seen as an invariant generalized determinant with a series representation indexed by any elliptical generator function. In particular, a corollary emerges for a simple derivation of the central matrix variate beta type II distribution under elliptically contoured models in the unified real, complex, quaternions and octonions.

翻译：本文在椭圆轮廓模型与实赋范可除代数框架下，提出了标量二项级数的矩阵扩展。经典超几何级数 ${}_{1}F_{0}^{\beta}(a;\mathbf{Z})={}_{1}^{k}P_{0}^{\beta,1}(1:a;\mathbf{Z})=|\mathbf{I}-\mathbf{Z}|^{-a}$ 所对应的Jack多项式，现可视为一种具有不变性的广义行列式，其级数表示可由任意椭圆生成函数索引。特别地，本文推导出一个推论，为在统一的实数、复数、四元数与八元数域中，椭圆轮廓模型下的中心矩阵变量II型Beta分布提供了简洁的推导方法。